AI 中文总结
该研究引入原子smashing标架推广Balmer谱，建立smashing理想与局部刚性局部化的对应，在稳定/非稳定情形下刻画广义望远镜猜想，并应用于色同伦论与连通模范畴研究。
AI 中文摘要
我们引入原子smashing frame（smashing标架），将张量三角几何中的Balmer谱（巴尔默谱）推广到任意可展示对称幺半∞-范畴$\u2118$中。由此给出了$\u2118$上望远镜猜想的一种表述，并在稳定紧刚性生成情形下复现了经典的Balmer谱。\n 利用可对偶与刚性∞-范畴，我们建立了smashing理想与局部刚性局部化之间的对应关系。由此得到（预）稳定情形下smashing标架的一个recollement（粘合）定理，以及稳定紧刚性生成情形下的原子精细化结果。作为在色同伦论中的核心应用，我们证明自然投影诱导了$\text{Sp}$（谱范畴）的smashing标架到所有素数与高度下单色层$\text{Sp}_{T(n)}$的smashing标架的乘积中的嵌入。特别地，$\text{Sp}$的smashing标架的空间性完全约化为$\text{Sp}_{T(n)}$的smashing标架的空间性。\n 在非稳定情形下，我们用smashing field（smashing域）刻画了∞-topos（∞-意象）的望远镜猜想，并用Pierce型条件刻画了连通模范畴与超完全连通层的望远镜猜想。最后，我们引入了Serre smashing frame（塞尔smashing标架）。在连通$\u2102_\u221e$-环$R$上，该标架介于原子smashing标架与通常的smashing标架之间，为连通$R$-模范畴的望远镜猜想研究提供了一个中间结构层。
英文摘要
We extend Balmer's tensor-triangular geometry to arbitrary presentably symmetric monoidal $\infty$-categories $\mathcal V$ by introducing the atomic smashing frame. This construction is functorial in $\mathcal V$, recovers the classical Balmer spectrum in the stable compactly-rigidly generated case, and yields a formulation of the telescope conjecture for arbitrary $\mathcal V$. Using dualizable and locally rigid $\infty$-categories, we identify smashing ideals with locally rigid localizations and establish recollement results for smashing frames and atomic smashing frames. As a consequence, we obtain a recollement principle for the telescope conjecture. We also establish descent for the telescope conjecture along several natural classes of coverings. As a principal application to chromatic homotopy theory, we analyze the structure of the smashing frame of $\mathrm{Sp}$. We prove a prime decomposition in terms of the smashing frames of the $p$-local $\infty$-categories $\mathrm{Sp}_{(p)}$ and, at each prime, an embedding into the product of the smashing frames of the monochromatic layers $\mathrm{Sp}_{T(n)}$. In particular, the spatiality of the smashing frame of $\mathrm{Sp}$ reduces entirely to that of $\mathrm{Sp}_{T(n)}$. Beyond the stable setting, we characterize the telescope conjecture for $\infty$-topoi in terms of smashing fields, and for connective module categories and hypercomplete connective sheaves in terms of Pierce-type conditions. Finally, we introduce the Serre smashing frame. Over a connective $\mathbb{E}_\infty$-ring $R$, this frame sits between the atomic and usual smashing frames, providing an intermediate structural layer in the study of the telescope conjecture for the connective $R$-module $\infty$-category.
Commentsv2: Added a dualizable internal-Hom formula for the Artin gluing and a recollement principle for the telescope conjecture; slightly reorganized the exposition
